Search: id:A050292
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%I A050292
%S A050292 1,1,2,3,4,4,5,5,6,6,7,8,9,9,10,11,12,12,13,14,15,15,16,16,17,17,18,19,
%T A050292 20,20,21,21,22,22,23,24,25,25,26,26,27,27,28,29,30,30,31,32,33,33,34,
%U A050292 35,36,36,37,37,38,38,39,40,41,41,42,43,44,44,45,46,47,47,48,48,49,49,
               50,51,52,52,53,54
%N A050292 Maximal cardinality of a double-free subset of {1, 2, ..., n}.
%C A050292 Maximal size of a subset S of {1, 2, ..., n} with the property that if 
               x is in S then 2x is not.
%C A050292 Least k such that a(k)=n is equal to A003159(n).
%C A050292 To construct the sequence : let [a, b, c, a, a, a, b, c, a, b, c, ...] 
               the fixed point of the morphism a -> abc, b ->a, c -> a, starting 
               from a(1) = a, then write the indices of a, b, c that of a being 
               written twice; see A092606 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Apr 13 2004
%D A050292 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.26.
%D A050292 Wang, E. T. H. ``On Double-Free Sets of Integers.'' Ars Combin. 28, 97-100, 
               1989.
%H A050292 S. R. Finch, <a href="http://algo.inria.fr/csolve/triple/">Triple-Free 
               Sets of Integers</a>
%H A050292 R. Stephan, <a href="somedcgf.html">Some divide-and-conquer sequences 
               ...</a>
%H A050292 R. Stephan, <a href="a079944.ps">Table of generating functions</a>
%H A050292 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Double-FreeSet.html">Link to a section of The World of Mathematics.</
               a>
%F A050292 Partial sums of A035263. Close to (2/3)*n.
%F A050292 a(1)=1, a(n)=n-a(floor(n/2)); a(n)=(2/3)*n+(1/3)*A065359(n); more generally, 
               for m>=0, a(2^m*n)-2^m*a(n)=A001045(m)*A065359(n) where A001045(m)={2^m-(-1)^m}/
               3 is the Jacobsthal sequence; a(A039004(n))=(2/3)*A039004(n); a(2*A039004(n))=2*a(A039004(n)); 
               a(A003159(n))=n; a(A003159(n)-1)=n-1; a(n)(mod 2)=A010060(n) the 
               Thue-Morse sequence; a(n+1)-a(n)=A035263(n+1); a(n+2)-a(n) = abs(A029884(n)). 
               - Benoit Cloitre, Nov 24, 2002
%F A050292 Series expansion: (1/(x*(x-1))) * Sum(i=0, infinity, (-1)^i*x^(2^i)/(x^(2^i)-1) 
               ). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 
               2003
%F A050292 a(n)=sum(k=>0, (-1)^k*floor(n/2^k)) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Jun 03 2003
%F A050292 a(A091785(n)) = 2n; a(A091855(n)) = 2n-1 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Mar 26 2004
%e A050292 Examples for n = 1 through 8: {1}, {1}, {1,3}, {1,3,4}, {1,3,4,5}, {1,
               3,4,5}, {1,3,4,5,7}, {1,3,4,5,7}.
%t A050292 a[n_] := a[n] = If[n < 2, 1, n - a[Floor[n/2]]]; Table[ a[n], {n, 1, 
               75}]
%o A050292 (PARI) a(n)=if(n<2,1,n-a(floor(n/2)))
%Y A050292 Cf. A050291-A050296, A050321, A035263.
%Y A050292 Cf. A121701.
%Y A050292 Sequence in context: A047741 A047784 A047742 this_sequence A071521 A039733 
               A005374
%Y A050292 Adjacent sequences: A050289 A050290 A050291 this_sequence A050293 A050294 
               A050295
%K A050292 nonn,nice,easy
%O A050292 1,3
%A A050292 Eric Weisstein (eric(AT)weisstein.com)
%E A050292 Extended with formula by Christian G. Bower (bowerc(AT)usa.net), Sep 
               15 1999.
%E A050292 Corrected and extended by Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Aug 16 2006

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